scholarly journals Containment Game Played on Random Graphs: Another Zig-Zag Theorem

10.37236/4777 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Paweł Prałat
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Domination Number ◽  
Main Question ◽  
Wide Range ◽  
Minimum Number ◽  
Natural Counterpart ◽  
Delta G ◽  
The Moment ◽  
Zigzag Shape

We consider a variant of the game of Cops and Robbers, called Containment, in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop). The cops win by "containing'' the robber, that is, by occupying all edges incident with a vertex occupied by the robber. The minimum number of cops, $\xi(G)$, required to contain a robber played on a graph $G$ is called the containability number, a natural counterpart of the well-known cop number $c(G)$. This variant of the game was recently introduced by Komarov and Mackey, who proved that for every graph $G$, $c(G) \le \xi(G) \le \gamma(G) \Delta(G)$, where $\gamma(G)$ and $\Delta(G)$ are the domination number and the maximum degree of $G$, respectively. They conjecture that an upper bound can be improved and, in fact, $\xi(G) \le c(G) \Delta(G)$. (Observe that, trivially, $c(G) \le \gamma(G)$.) This seems to be the main question for this game at the moment. By investigating expansion properties, we provide asymptotically almost sure bounds on the containability number of binomial random graphs $\mathcal{G}(n,p)$ for a wide range of $p=p(n)$, showing that it forms an intriguing zigzag shape. This result also proves that the conjecture holds for some range of $p$ (or holds up to a constant or an $O(\log n)$ multiplicative factors for some other ranges).

scholarly journals Metric Dimension for Random Graphs

10.37236/2639 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Béla Bollobás ◽  
Dieter Mitsche ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Metric Dimension ◽  
Wide Range ◽  
Minimum Number ◽  
Vertex Set

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

scholarly journals Independent Domination Subdivision in Graphs

2021 ◽  
Author(s):  
Ammar Babikir ◽  
Magda Dettlaff ◽  
Michael A. Henning ◽  
Magdalena Lemańska
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Independent Domination ◽  
Delta G ◽  
Independent Dominating Set ◽  
Domination Subdivision Number

AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$ sd γ ( G ) . For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$ sd γ ( G ) ≤ 3 , while $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$ Δ ( G ) such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$ sd i ( G ) ≥ 3 Δ ( G ) - 2 , in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$ sd γ ( G ) ≤ 2 Δ ( G ) - 1 always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$ sd i ( T ) = 1 .

scholarly journals The Chromatic Index of a Graph Whose Core has Maximum Degree $2$

10.37236/2101 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Mikio Kano ◽  
Saieed Akbari ◽  
Maryam Ghanbari ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Chromatic Index ◽  
The Core ◽  
Class 1 ◽  
Minimum Number ◽  
Even Order ◽  
Delta G

Let $G$ be a graph. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by the vertices of degree $\Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. A $k$-edge coloring of $G$ is a function $f:E(G)\rightarrow L$ such that $|L| = k$ and $f(e_1)\neq f(e_2)$ for all two adjacent edges  $e_1$ and $e_2$ of $G$. The chromatic index of $G$, denoted by $\chi'(G)$, is the minimum number $k$ for which $G$ has a $k$-edge coloring.  A graph $G$ is said to be Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if $\chi'(G) = \Delta(G) + 1$. In this paper it is shown that every connected graph $G$ of even order and with $\Delta(G_{\Delta})\leq 2$ is Class $1$ if $|G_{\Delta}|\leq 9$ or $G_{\Delta}$ is a cycle of order $10$.

scholarly journals Bounds for Distinguishing Invariants of Infinite Graphs

10.37236/6362 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Wilfried Imrich ◽  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mohammad Hadi Shekarriz
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Infinite Graphs ◽  
Chromatic Index ◽  
Distinguishing Number ◽  
Minimum Number ◽  
Edge Colourings ◽  
Delta G ◽  
Vertex Colouring

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.

scholarly journals On Asymmetric Colourings of Claw-Free Graphs

10.37236/8886 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Wilfried Imrich ◽  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Mariusz Woźniak
Keyword(s):  
Maximum Degree ◽  
Distinguishing Number ◽  
Identity Automorphism ◽  
Minimum Number ◽  
Large Motion ◽  
Delta G ◽  
Vertex Colouring

A vertex colouring of a graph is asymmetric if it is preserved only by the identity automorphism. The minimum number of colours needed for an asymmetric colouring of a graph $G$ is called the asymmetric colouring number or distinguishing number $D(G)$ of $G$. It is well known that $D(G)$ is closely related to the least number of vertices moved by any non-identity automorphism, the so-called motion $m(G)$ of $G$. Large motion is usually correlated with small $D(G)$. Recently, Babai posed the question whether there exists a function $f(d)$ such that every connected, countable graph $G$ with maximum degree $\Delta(G)\leq d$ and motion $m(G)>f(d)$ has an asymmetric $2$-colouring, with at most finitely many exceptions for every degree. We prove the following result: if $G$ is a connected, countable graph of maximum degree at most 4, without an induced claw $K_{1,3}$, then $D(G)= 2$ whenever $m(G)>2$, with three exceptional small graphs. This answers the question of Babai for $d=4$ in the class of~claw-free graphs.

scholarly journals Random Lifts of Graphs are Highly Connected

10.37236/2783 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Marcin Witkowski
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Asymptotic Properties ◽  
New Model ◽  
Base Graph ◽  
The Core ◽  
Similar Idea ◽  
Delta G

In this note we study asymptotic properties of random lifts of graphs introduced by Amit and Linial as a new model of random graphs. Given a base graph $G$ and an integer $n$, a random lift of $G$ is obtained by replacing each vertex of $G$ by a set of $n$ vertices, and joining these sets by random matchings whenever the corresponding vertices of $G$ are adjacent. In this paper we study connectivity properties of random lifts. We show that the size of the largest topological clique in typical random lifts, with $G$ fixed and $n\rightarrow\infty$, is equal to the maximum degree of the core of $G$ plus one. A similar idea can be used to prove that for any graph $G$ with $\delta(G)\geq2k-1$ almost every random lift of $G$ is $k$-linked.

ON PALETTE INDEX OF UNICYCLE AND BICYCLE GRAPHS

2019 ◽  
Vol 53 (1 (248)) ◽  
pp. 3-12
Author(s):  
A.B. Ghazaryan
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Cyclomatic Number ◽  
Proper Edge Coloring ◽  
Minimum Number ◽  
Delta G

Given a proper edge coloring $ \phi $ of a graph $ G $, we define the palette $ S_G (\nu, \phi) $ of a vertex $ \nu \mathclose{\in} V(G) $ as the set of all colors appearing on edges incident with $ \nu $. The palette index $ \check{s} (G) $ of $ G $ is the minimum number of distinct palettes occurring in a proper edge coloring of $ G $. In this paper we give an upper bound on the palette index of a graph G in terms of cyclomatic number $ cyc(G) $ of $ G $ and maximum degree $ \Delta (G) $ of $ G $. We also give a sharp upper bound for the palette index of unicycle and bicycle graphs.

Industrial Labour Force of Russia in 20-30s of XX Century in Russian Historiography

2020 ◽  
pp. 431-449
Author(s):  
Oleg V. Shekatunov ◽  
Konstantin G. Malykhin
Keyword(s):  
Labour Force ◽  
Historical Period ◽  
Russian Industry ◽  
Wide Range ◽  
Scientific Papers ◽  
Potential Formation ◽  
The Moment ◽  
Force Potential ◽  
Modern Russian

The article is devoted to the specifics of studying the industrial labour force of Russia in the 1920s - 1930s in Russian historiography. The various stages of study from the 1920s through the 1930s and up to the last years are concerned. The relevance of the study is due to several factors. These include contradictions in the assessments of Bolshevik modernization of the 1920s and 1930s; projected labour force shortages in modern Russia; as well as the existing labour force shortage in industry at the moment. This determines the relevance of studying the historical period, which was characterized by the most acute personnel problems in the country. The novelty of the study is due to the fact that in modern Russian historiography there is no holistic, integrated view of the problems of the labour force potential formation of Russian industry in the 1920s and 1930s. It is noted that there is no research aimed at analyzing the historiography of these problems. The main stages of the study of industrial labour force are highlighted. The analysis of scientific works correlated with each stage of the study of the topic is performed. The problems and methodology of each stage are considered. A review of a wide range of scientific papers both articles and thesis is presented.

Potential Triazole-based Molecules for the Treatment of Neglected Diseases

2019 ◽  
Vol 26 (23) ◽  
pp. 4403-4434 ◽  
Author(s):  
Susimaire Pedersoli Mantoani ◽  
Peterson de Andrade ◽  
Talita Perez Cantuaria Chierrito ◽  
Andreza Silva Figueredo ◽  
Ivone Carvalho
Keyword(s):  
Effective Treatment ◽  
Chagas Disease ◽  
Medicinal Chemistry ◽  
Biological Activities ◽  
Neglected Diseases ◽  
Triazole Derivatives ◽  
Wide Range ◽  
The World ◽  
Great Relevance ◽  
The Moment

Neglected Diseases (NDs) affect million of people, especially the poorest population around the world. Several efforts to an effective treatment have proved insufficient at the moment. In this context, triazole derivatives have shown great relevance in medicinal chemistry due to a wide range of biological activities. This review aims to describe some of the most relevant and recent research focused on 1,2,3- and 1,2,4-triazolebased molecules targeting four expressive NDs: Chagas disease, Malaria, Tuberculosis and Leishmaniasis.

scholarly journals EXPRESS: Overcoming the Cold Start Problem of CRM using a Probabilistic Machine Learning Approach

2021 ◽  
pp. 002224372110329
Author(s):  
Nicolas Padilla ◽  
Eva Ascarza
Keyword(s):  
Machine Learning ◽  
Cold Start ◽  
Customer Relationship ◽  
Exponential Families ◽  
Modeling Framework ◽  
Wide Range ◽  
The Moment ◽  
Probabilistic Machine Learning ◽  
Marketing Actions ◽  
Cold Start Problem

The success of Customer Relationship Management (CRM) programs ultimately depends on the firm's ability to identify and leverage differences across customers — a very diffcult task when firms attempt to manage new customers, for whom only the first purchase has been observed. For those customers, the lack of repeated observations poses a structural challenge to inferring unobserved differences across them. This is what we call the “cold start” problem of CRM, whereby companies have difficulties leveraging existing data when they attempt to make inferences about customers at the beginning of their relationship. We propose a solution to the cold start problem by developing a probabilistic machine learning modeling framework that leverages the information collected at the moment of acquisition. The main aspect of the model is that it exibly captures latent dimensions that govern the behaviors observed at acquisition as well as future propensities to buy and to respond to marketing actions using deep exponential families. The model can be integrated with a variety of demand specifications and is exible enough to capture a wide range of heterogeneity structures. We validate our approach in a retail context and empirically demonstrate the model's ability at identifying high-value customers as well as those most sensitive to marketing actions, right after their first purchase.

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